A Test for the Number of Factors in an Approximate Factor Model

Abstract

An important issue in applications of multifactor models of asset returns is the appropriate number of factors.Most extant tests for the number of factors are valid only for strict factor models, in which diversifiable returns are uncorrelated across assets.In this paper we develop a test statistic to determine the number of factors in an approximate factor model of asset returns, which does not require that diversifiable components of returns be uncorrelated across assets.We find evidence for one to six pervasive factors in the cross-section of New York Stock Exchange and American Stock Exchange stock returns.1976) has generated an increased interest in the application of linear factor models in the study of capital asset pricing.The APT has the attractive feature that it makes a minimal number of assumptions about the nature of the economy (a factor structure for the returns generating process, a large number of assets, and frictionless trading).The costs of these minimalist assumptions include certain ambiguities such as an approximate pricing relation and an unknown number of pervasive factors. THE ARBITRAGE PRICING THEORY (APT) of Ross (In order to estimate and test the APT, one must specify the number of pervasive factors in asset returns.The issue of the appropriate number of factors has been the subject of some controversy (see, for example, Roll and Ross (1980, 1984); Dhrymes, Friend, and Gultekin (1984); Luedecke (1984); Trzcinka (1986); Conway and Reinganum (1988); and Brown (1989)).In this paper we propose a new approach to estimating the number of pervasive economic factors generating asset returns.An important feature of our approach is that it is valid when asset returns follow an approximate, rather than a strict, factor model.A strict factor structure is one in which the idiosyncratic, or diversifiable, components of asset returns have zero correlation across assets.Ross (1976) assumes a strict factor structure in his original development of the APT.However, he notes that this assumption can be weakened.The key requirement for the APT is that nonfactor risk can be diversified away in many-asset